Advancing and retreating fronts in a changing climate: a percolation model of range shifts

Climate change causes considerable shifts in the geographic distribution of species worldwide. Most data on range movements, however, derive from relatively short periods, within which it is difficult to distinguish directional shifts from random fluctuations. For detecting a shift, it is indispensable to delineate the range precisely. We propose a new method for the delineation based on percolation theory. We suggest marking the boundary between the connected and fragmented occurrence of the species (the hull). We demonstrate the advantages of this connectivity-based method on simulated examples in which a metapopulation is advancing vs retreating along an environmental gradient with different velocities. The simulations show that the hull is a fractal and has the same dimension (7/4) even when the front is advancing or retreating relatively fast, compared to the generation time. It is particularly robust in the retreating (trailing) edge. Accordingly, we propose marking the range edge at the mean position of the hull, the 'connectivity limit' of the species. Theoretical considerations suggest that the position of the connectivity limit is statistically more reliable than those limits that are delineated according to the outermost occurrences, and the connectivity-based method is broadly applicable to real-life data.